Prior art solutions to inventory management have focused on total cost and customer service. For example, Goll et al. (U.S. Pat. App. No. 2006/0085299 A1) provide a method and system to manage inventory based on order quantity and safety stock quantity such that total cost is minimized, while maintaining a desired customer service level. Ettl et al. (U.S. Pat. No. 5,946,662) provide a method to manage inventory levels for products in a complex supply chain network based on total inventory cost and customer fill rates. Koray et al. (U.S. Pat. App. No. 2004/0230475) provide a system to optimize inventory targets for nodes of a supply chain to satisfy a target customer service level.
Many formulas and algorithms have been created to minimize total cost. Fundamental to these methods is the Economic Order Quantity (EOQ), or Lot Size, Model. Still in use today, the model was originally developed by F. W. Harris in 1913. For the history of the EOQ Model, see D. Erlenkotter, “Ford Whitman Harris and the Economic Order Quantity Model”, 38 Operations Research, 937-946 (1990). The formulation and operation of the model are explained by E. A. Silver, D. F. Pyke, and R. Peterson, Inventory Management and Production Planning and Scheduling, Wiley (3rd ed. 1998), 784 pp., pages 149-155.
We will summarize the EOQ Model for comparison with the present invention. For convenience in the discussion that follows, time will be expressed in years (yr). Currency will be expressed in dollars ($). Quantity will be expressed in units; in some contexts, fractional units make sense (e.g., bushels of wheat), but in others they do not (e.g., number of laptop computers). Unless otherwise specified in context, discussion of the present invention pertains to either situation.
The order quantity, or run size, Q (in units) represents a quantity of a particular item ordered from a supplier or facility, such as a distribution center or a manufacturer. Suppliers may be external vendors, manufacturers, distributors, warehouses, or any other entity or facility capable of supplying goods. The EOQ Model recommends ordering the quantity Qo that minimizes the total variable cost required to order and hold inventory. Determination of Qo involves the following four basic variables:                D: the demand (in units/yr). D is the annual unit demand forecast, which represents the need for a particular product or component. The demand could come from any number of sources, for example, customer orders, forecasts, interplant requirements, or requests from a branch warehouse for service parts or raw materials for manufacturing. Customers may be individuals, middlemen, or facilities acting like a customer in a supply chain (e.g. a manufacturer, distributor, warehouse, and the like). At the finished goods level, “demand data” are usually different from “sales data” because demand does not necessarily result in sales; for example, if there is no stock, there will be no sale.        Co: the ordering cost (in $). Co is the fixed portion of the cost for placing and setting up a single order, and is therefore independent of Q. It includes those costs that increase as the number of orders placed increases. It includes costs related to the clerical work of preparing, releasing, following, and receiving orders; that portion of the physical handling of goods which is not dependent on quantity; inspection; and setup costs, as applicable. In today's environment of increasing fuel costs, Co also includes transportation costs.        Cm: the unit cost (in $). Cm is the cost to buy an item of inventory (in the case of a retailer) or the cost of a component or a unit amount of raw material (in the case of a manufacturer), plus the marginal costs of value-added processing.        rh: the holding cost ratio (in yr−1). rh is the cost of holding one dollar of inventory for one year, and is usually expressed as a percentage of Cm per year. rh includes storage, obsolescence, shrinkage, property insurance coverage, property taxes, and cost of capital. A commonly assumed value is 0.225/yr.        
The total variable cost of ordering and holding inventory (in $/yr) is given byĈ=Ĉo+Ĉh  (1)where the total ordering cost isĈo=CoD/Q.  (2)and the total holding cost isĈh=Cmrh,  (3)where  is the average inventory;  will be discussed below.
The customer service level λ (dimensionless) is the ratio of the number of orders completed on time to the number of orders placed. For example, λ=0.97 means that 97% of orders can be filled immediately from available stock. The safety stock quantity Z (in units) is inventory in excess of forecast demand that is kept on hand to avoid stockouts and to maintain a high value of λ.
Two variations of the formula for Ĉh, and correspondingly two variations in the calculation of EOQ, will be described below, the two variations differing as to whether safety stock is taken into account. If safety stock is not considered, determination of a closed-form expression for the economic order quantity Qo is possible. In this case,=Q/2  (4)so the total holding cost (in $/yr) isĈh=CmrhQ/2.  (5)
Substituting equations (2) and (5) into (1), differentiating with respect to Q, setting the result equal to zero, and solving for Q yields the EOQ for the case where safety stock is not considered,
                              Q          o                =                                            (                                                2                  ⁢                                      C                    o                                    ⁢                  D                                                                      C                    m                                    ⁢                                      r                    h                                                              )                                      1              /              2                                .                                    (        6        )            
FIG. 1 illustrates a specific example of the EOQ Model, for subsequent comparison with the approach of the present invention. In this example, the parameters are D=1200 unit/yr, Cm=$50, Co=$100, and rh=0.23/yr. The vertical axis 100 in the figure is cost. The horizontal axis 110 is order quantity Q. The dependence of the total annual variable cost of ordering and holding inventory Ĉo upon Q is shown in the figure with a dotted line 120. The shape of this curve reflects the fact that the number of orders within a year decreases as Q increases. The dependence of total annual holding cost Ĉh upon Q is shown with a dashed line 130. Ĉh increases linearly as Q increases. The sum of these two curves Ĉ, the total cost, is shown with a solid line 140. The minimum 150 of the Ĉh curve 130 occurs at the order quantity Q=Qo indicated by reference number 160, which corresponds to the minimum total cost Ĉh indicated by reference number 170. In this particular example in which safety stock has been omitted from the calculation, the value of Qo is 144 units, as expected from equation (6).
A more realistic formulation of EOQ incorporates safety stock quantity Z and variability in lead time τ. τ (in yr) is the time interval between when an order is placed from a supplier and when the ordered goods are received. In this case, determination of an explicit formula for Qo analogous to equation (6) is not possible. Qo can, however, be determined by numerical solution.
A widely-used formula for safety stock quantity (Silver et al., p. 244) isZ=kσ  (7)where k is called the safety factor and σ is the standard deviation of the combined variability of demand during the forecast replenishment lead time and the variability of demand due to deviation in the lead time from the forecast lead time:σ=√{square root over ( τσD2+( Dστ)2)}  (8)where D and σD (in units) are, respectively, the mean and standard deviation of demand during the replenishment lead time, each multiplied by some characteristic time scale τ0. τ and στ are the mean and standard deviation of the lead time, each nondimensionalized by the time scale τ0. Equation (8) is discussed by T. E. Vollmann et al., Manufacturing Planning and Control Systems for Supply Chain Management, (McGraw-Hill, 5th ed. 2005), 712 pp., pages 133-135.
Additional safety stock should be carried if the actual replenishment quantity can vary from what was ordered. In this case, the following equation is used in lieu of (8):σ=√{square root over ( τσD2+( Dστ)2+σQ2)},  (9)where σQ is the standard deviation of the replenishment quantity (in units) over the interval of time τ0.
Higher values of the safety factor k correspond to higher customer service levels (λ). Software tools are commercially available to determine k. Inventory planning tools are available in the SAP APO, for example, allowing safety factor k to be determined based on a service level over the lead time. Some packages, including MathCAD and Microsoft, facilitate empirical calculation of a formulation of k based on statistics the observed service level over lead time within the particular company engaged in the inventory management process.
Silver et al. (p. 736) provide the following formula for k:
                    k        =                                            a              0                        +                                          a                1                            ⁢              x                        +                                          a                2                            ⁢                              x                2                                      +                                          a                3                            ⁢                              x                3                                                                        b              0                        +                                          b                1                            ⁢              x                        +                                          b                2                            ⁢                              x                2                                      +                                          b                3                            ⁢                              x                3                                      +                                          b                4                            ⁢                              x                4                                                                        (        10        )            where x=√{square root over (ln(25/δ2(k)))} andδ(k)=(1−λ)Q/σ.  (11)Note that equation (11) implies that Q depends σ.
The coefficients in equation (8) are
a0−5.3925569b01a15.6211054b1 −7.2496485 × 10−1a2−3.8836830b2  5.07326622 × 10−1a31.0897299b3  6.69136868 × 10−2b4−3.29129114 × 10−3.
When safety stock is included in the model, the average inventory is=Q/2+Z  (12)The total holding cost becomesĈh=(Q/2+Z)Cmrh  (13)
Substituting equations (2) and (13) into (1) gives:Ĉ=CoD/Q+(Q/2+Z)Cmrh  (14)Analogously to the previous case in which storage stock was ignored, differentiating (1) with respect to Q, and setting the result equal to zero yields an equation for the economic order quantity Qo. Although this equation cannot be solved in closed-form, a value of Qo can be determined for any given combination of parameters using standard numerical equation-solving techniques well known to persons of ordinary skill in the art, such as those found in W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C, 1992, 994 pp. Alternatively, well known minimization techniques can be applied directly to find the value of Q that minimizes Ĉ in (14); several such techniques are also provided by Press et al.
One of ordinary skill in the art will also recognize that software packages used to determine safety stock quantity based on lead time are known, such as SAP Advanced Planner and Optimization, available from SAP AG (Walldorf, Germany). Furthermore, software packages used to optimize inventory levels, optimize supply chain design, and to optimize supply chain plans relative cost and time are known, such as INVENTORY ANALYST available from LogicTools, Inc., which performs multi-echelon inventory optimization. INVENTORY ANALYST is part of the integrated tool SUPPLY CHAIN ANALYST, which also includes tools for distribution-focused analysis, seasonal build, sales and operations, demand planning, product flow optimization, and supply planning.
FIG. 2 illustrates the case just described in which total cost Ĉ incorporates the effect of safety stock, but is otherwise analogous to FIG. 1. The values of the parameters (D=1200 unit/yr, Cm=$50, Co=$100, and rh=0.23/yr) are the same as in the previous example. Additional parameters used in the calculation include: λ=0.97, τ=28 days, στ=5 days, D=23 units/wk, and σD=10 units/wk. (Note that all these times must be first converted to years for the equations above to be used directly.) The vertical axis 200, the horizontal axis 210, the Ĉo curve 220, the Ĉh curve 230, and the Ĉ curve 240 should be interpreted in the same way as their counterparts in FIG. 1. However, here the Ĉh curve 230 includes the cost of safety stock, the contribution of which is shown in a separate curve 280. As in FIG. 1, the minimum 250 of the Ĉh curve 230 occurs at the order quantity Q=Qo indicated by reference number 260, which corresponds to the minimum total cost, indicated by reference number 270. Inclusion of safety stock results in a recommended order quantity Qo having a value (rounded to the nearest whole number) of 162 units, an increase of 18 units from the case described previously in which safety stock was ignored.